Eulers Product Exhaustion Theorem

Meaning and Solution

The product exhaustion
theorem states that since factors of production are rewarded equal to their marginal
product, they will exhaust the total product. The way this proposition is solved has
been called the adding up problem. Wick steed in The Coordination of the Laws of Distribution
demonstrated with the help of Euler’s Theorem, that payment in accordance with
marginal productivity to each factor exactly exhausts the total product.

The adding up problem
states that in a competitive factor market when every factor employed in the production
process is paid equal to the value of its marginal product, then pay in the production
process is paid a price equal to the value of its marginal product, then payments to
the factors exhaust the total value of the product. It can be represented numerically
as under:

Q = (MPL) L + (MPc)
C

Where, Q is total output, MP is marginal product, L is labour and K is capital. To find
out the value of output, multiply through P (Price). Thus

P x Q = (MPL X P)
L + (MPc x P) C

(MPL X P) = VMPL
and (MPc x P) = VMPc

PQ = VMPL x VMPc

Where, VMPL is the value of marginal product of labour and VMPc is the value of marginal
product of capital.

Euler’s Product Exhaustion Theorem

Postulations

It assumes a linear standardised production of first degree which implies invariable
returns to scale.

It assumes that the factors are complementary, i.e. if a variable factor increases;
it increases the marginal productivity of the fixed factor.

It assumes that factors of production are perfectly divisible.

The relative shares of the factors are invariable and independent of the level
of the product.

There is a stationary, reckless economy where there are no profits.

There is perfect competition.

It is applicable only in the long run.

Explanation

Based on these postulations
of Euler, Wicksteed proved his theorem that when each factor was paid according to
its marginal product, the total product would be exactly exhausted. This is based on
the postulation of a linear standardised function. Few economists criticised his work
and pointed out that the production function does not yield a horizontal long run average
cost curve LRAC but a U Shaped LRAC curve. The U shaped LRAC curve first shows decreasing
returns to scale, then constant and in the end increasing returns to scale.

The solution of the
product exhaustion theorem is based on a profitless long run, perfectly competitive
equilibrium position of a industry which operates at the minimum point, E of its LRAC
curve as represented in the Diagram (1).

At this point the firm is in full equilibrium, the marginal revenue productivity MRP
of the factors being equal to the combined marginal cost of the factors MFC. This is
represented in the Diagram 2.

Where, MRP = MFC at point A. It is at point A that the total product OQ is exactly distributed
to OM factors and nothing is left over.

The product exhaustion problem is solved with a linear standardised production function:

P
= δ P C + δ P L
δC δL

Nevertheless there are diminishing returns to scale, less than the total product will
be paid to the factors:

P > δ P C + δ P L
δC δL

In such a condition, there will be super normal profits in the industry. They will attract
new firms into the industry. Consequently output will increase, price will fall and
profits will be eradicated in the long run. In this way, the distributive shares of
the factors as determined by their marginal productivities will absolutely exhaust
the total product.

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